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Operator corona problems for function algebras
Ryan Hamilton
Pure Math. Dept.U. Waterloo
Waterloo, ON
August 2011BanAlg 2011, University of Waterloo
Joint work with M. Raghupathi (U.S. Naval Academy)
Ryan Hamilton Operator corona problems for function algebras
Ryan Hamilton Operator corona problems for function algebras
Carleson’s corona theorem
Let H∞ denote the bounded, analytic functions on the unit disk D.
Theorem (Carleson 1962)
Suppose f1, . . . , fn ∈ H∞ satisfy
n∑i=1
|fi (z)|2 ≥ δ2 > 0, z ∈ D.
Then there are functions g1, . . . , gn ∈ H∞ such that∑n
i=1 figi = 1.
Ryan Hamilton Operator corona problems for function algebras
The Toeplitz corona theorem
Theorem (Arveson-1975; Schubert-1978)
Suppose f1, . . . , fn ∈ H∞ satisfy
n∑i=1
TfiT∗fi≥ c2I .
Then there are functions g1, . . . , gn ∈ H∞ so that
n∑i=1
figi = 1, and ‖[Tgi , . . . ,Tgn ]T‖ ≤ c−1.
Ryan Hamilton Operator corona problems for function algebras
Operator corona problems
Suppose A is an operator algebra and A1, . . . ,An ∈ A with[A1, . . . ,An] right invertible.
There are B1, . . . ,Bn ∈ B(H) such that [A1, . . . ,An]
B1...Bn
= IH .
For example, take Bi = A∗i (∑n
i=1 AiA∗i )−1.
Can we find Bi in A?
Ryan Hamilton Operator corona problems for function algebras
Operator corona problems
Suppose A is an operator algebra and A1, . . . ,An ∈ A with[A1, . . . ,An] right invertible.
There are B1, . . . ,Bn ∈ B(H) such that [A1, . . . ,An]
B1...Bn
= IH .
For example, take Bi = A∗i (∑n
i=1 AiA∗i )−1.
Can we find Bi in A?
Ryan Hamilton Operator corona problems for function algebras
Operator corona problems
Suppose A is an operator algebra and A1, . . . ,An ∈ A with[A1, . . . ,An] right invertible.
There are B1, . . . ,Bn ∈ B(H) such that [A1, . . . ,An]
B1...Bn
= IH .
For example, take Bi = A∗i (∑n
i=1 AiA∗i )−1.
Can we find Bi in A?
Ryan Hamilton Operator corona problems for function algebras
Operator corona theorem for nest algebras
Suppose Pmm≥0 is an increasing sequence of projections tendingstrongly to IH and let A := AlgPmm≥0 .
Theorem (Arveson-1975)
Suppose A1, . . . ,An ∈ A satisfy
n∑k=1
AkPmA∗k ≥ c2Pm for every m ≥ 0.
Then there are B1, . . . ,Bn ∈ A such that
n∑k=1
AkBk = IH
Ryan Hamilton Operator corona problems for function algebras
Operator corona theorem for nest algebras
Suppose Pmm≥0 is an increasing sequence of projections tendingstrongly to IH and let A := AlgPmm≥0 .
Theorem (Arveson-1975)
Suppose A1, . . . ,An ∈ A satisfy
n∑k=1
AkPmA∗k ≥ c2Pm for every m ≥ 0.
Then there are B1, . . . ,Bn ∈ A such that
n∑k=1
AkBk = IH
Ryan Hamilton Operator corona problems for function algebras
Subalgebras of H∞
If B is an algebra of operators and h a vector, letB[h] := spanBh : B ∈ B.
Theorem (Raghupathi-Wick 2010)
Suppose A is a unital, weak∗-closed subalgebra of H∞ andf1, . . . , fn ∈ A satisfy
n∑i=1
TfiPLT∗fi≥ c2PL
for every L of the form A[h] where h is an outer function.Then there are g1, . . . , gn ∈ A so that
n∑i=1
figi = 1 and ‖[Tg1 , . . . ,Tgn ]T‖ ≤ c−1
Ryan Hamilton Operator corona problems for function algebras
Subalgebras of H∞
If B is an algebra of operators and h a vector, letB[h] := spanBh : B ∈ B.
Theorem (Raghupathi-Wick 2010)
Suppose A is a unital, weak∗-closed subalgebra of H∞ andf1, . . . , fn ∈ A satisfy
n∑i=1
TfiPLT∗fi≥ c2PL
for every L of the form A[h] where h is an outer function.Then there are g1, . . . , gn ∈ A so that
n∑i=1
figi = 1 and ‖[Tg1 , . . . ,Tgn ]T‖ ≤ c−1
Ryan Hamilton Operator corona problems for function algebras
The bidisk
Theorem (Amar 2003; Trent-Wick 2008)
Suppose f1, . . . , fn ∈ H∞(D2) satisfy
n∑i=1
T νfi
(T νfi
)∗ ≥ c2Iν
for every absolutely continuous measure ν on T2. Then there arefunctions g1, . . . , gn ∈ H∞(D2) so that
n∑i=1
figi = 1, and ‖[Tgi , . . . ,Tgn ]T‖ ≤ c−1.
Ryan Hamilton Operator corona problems for function algebras
Reproducing kernel Hilbert spaces
Let H be a Hilbert space of C-valued functions on a set X .If the functionals h 7→ h(x) are bounded, then we call H areproducing kernel Hilbert space (RKHS).
For each x ∈ X there is a function kx ∈ H such that
h(x) = 〈h, kx〉; h ∈ H.
The Hardy space H2 on the unit disk D is the canonical example.Its kernel is the Szego kernel
kw (z) =1
1− zw.
Ryan Hamilton Operator corona problems for function algebras
Reproducing kernel Hilbert spaces
Let H be a Hilbert space of C-valued functions on a set X .If the functionals h 7→ h(x) are bounded, then we call H areproducing kernel Hilbert space (RKHS).For each x ∈ X there is a function kx ∈ H such that
h(x) = 〈h, kx〉; h ∈ H.
The Hardy space H2 on the unit disk D is the canonical example.Its kernel is the Szego kernel
kw (z) =1
1− zw.
Ryan Hamilton Operator corona problems for function algebras
Reproducing kernel Hilbert spaces
Let H be a Hilbert space of C-valued functions on a set X .If the functionals h 7→ h(x) are bounded, then we call H areproducing kernel Hilbert space (RKHS).For each x ∈ X there is a function kx ∈ H such that
h(x) = 〈h, kx〉; h ∈ H.
The Hardy space H2 on the unit disk D is the canonical example.Its kernel is the Szego kernel
kw (z) =1
1− zw.
Ryan Hamilton Operator corona problems for function algebras
Examples of reproducing kernel Hilbert spaces
Let Ω ⊂ Cd be a bounded domain and let µ be Lesbesgue measureon Cn.
Example (Bergman space)
L2a(Ω) :=
f homomorphic on Ω :
∫Ω |f |
2dµ <∞
is a RKHS.
The kernel for L2a(D) is kw (z) = 1
(1−wz)2 .
Ryan Hamilton Operator corona problems for function algebras
Examples of reproducing kernel Hilbert spaces
Let Bd be the unit ball of Cd (we allow d =∞).
Example (Drury-Arveson space)
H2d is the closure of d-variable polynomials on Bd with kernel
kdw (z) =1
1− 〈z ,w〉Cd
H2d is an excellent multivariable analogue of H2
Many function spaces embed into H2∞ in a natural way
(Dirichlet space, Sobolev-Besov spaces).
Ryan Hamilton Operator corona problems for function algebras
Examples of reproducing kernel Hilbert spaces
Let Bd be the unit ball of Cd (we allow d =∞).
Example (Drury-Arveson space)
H2d is the closure of d-variable polynomials on Bd with kernel
kdw (z) =1
1− 〈z ,w〉Cd
H2d is an excellent multivariable analogue of H2
Many function spaces embed into H2∞ in a natural way
(Dirichlet space, Sobolev-Besov spaces).
Ryan Hamilton Operator corona problems for function algebras
Examples of reproducing kernel Hilbert spaces
Let Bd be the unit ball of Cd (we allow d =∞).
Example (Drury-Arveson space)
H2d is the closure of d-variable polynomials on Bd with kernel
kdw (z) =1
1− 〈z ,w〉Cd
H2d is an excellent multivariable analogue of H2
Many function spaces embed into H2∞ in a natural way
(Dirichlet space, Sobolev-Besov spaces).
Ryan Hamilton Operator corona problems for function algebras
Multiplier Algebras
The multiplier algebra of a RKHS H is
M(H) := f : X → C| fh ∈ H for any h ∈ H
For each multiplier f , the multiplication operator Mf ∈ B(H)defined by
Mf h = fh; h ∈ H
is automatically bounded by the closed graph theorem.We always have
M∗f kx = f (x)kx .
Ryan Hamilton Operator corona problems for function algebras
Multiplier Algebras
The multiplier algebra of a RKHS H is
M(H) := f : X → C| fh ∈ H for any h ∈ H
For each multiplier f , the multiplication operator Mf ∈ B(H)defined by
Mf h = fh; h ∈ H
is automatically bounded by the closed graph theorem.
We always haveM∗f kx = f (x)kx .
Ryan Hamilton Operator corona problems for function algebras
Multiplier Algebras
The multiplier algebra of a RKHS H is
M(H) := f : X → C| fh ∈ H for any h ∈ H
For each multiplier f , the multiplication operator Mf ∈ B(H)defined by
Mf h = fh; h ∈ H
is automatically bounded by the closed graph theorem.We always have
M∗f kx = f (x)kx .
Ryan Hamilton Operator corona problems for function algebras
Examples of multiplier algebras
M(H2((D))) = H∞(D)M(H2(D2)) = H∞(D2)M(L2
a(Ω)) = H∞(Ω)
M(H2d) ⊂ H∞(Bd) is the unital algebra generated by
multiplication by coordinates:
M(H2d) = Alg(Mz1 , . . . ,Mzd )
[Mz1 , . . . ,Mzd ] is the model for row contractions (Arveson1998).
Ryan Hamilton Operator corona problems for function algebras
Examples of multiplier algebras
M(H2((D))) = H∞(D)M(H2(D2)) = H∞(D2)M(L2
a(Ω)) = H∞(Ω)
M(H2d) ⊂ H∞(Bd) is the unital algebra generated by
multiplication by coordinates:
M(H2d) = Alg(Mz1 , . . . ,Mzd )
[Mz1 , . . . ,Mzd ] is the model for row contractions (Arveson1998).
Ryan Hamilton Operator corona problems for function algebras
The Toeplitz corona theorem for Drury-Arveson space
Theorem (Ball-Trent-Vinnikov 2002)
Suppose f1, . . . , fn ∈M(H2d) satisfy
n∑i=1
MfiM∗fi≥ c2I .
Then there are functions g1, . . . , gn ∈M(H2d) so that
n∑i=1
figi = 1.
and ‖[Mgi , . . . ,Mgn ]T‖ ≤ c−1.
Ryan Hamilton Operator corona problems for function algebras
Invariant subspaces
Suppose A is any unital, weakly closed algebra of multiplierson H.
Every invariant subspace L of A is also a RKHS, with kernelfunction kLλ := PLkλ.
For f ∈ A and g ∈ L we have fg ∈ L. Call this multiplicationoperator ML
f .
Ryan Hamilton Operator corona problems for function algebras
Invariant subspaces
Suppose A is any unital, weakly closed algebra of multiplierson H.
Every invariant subspace L of A is also a RKHS, with kernelfunction kLλ := PLkλ.
For f ∈ A and g ∈ L we have fg ∈ L. Call this multiplicationoperator ML
f .
Ryan Hamilton Operator corona problems for function algebras
Invariant subspaces
Suppose A is any unital, weakly closed algebra of multiplierson H.
Every invariant subspace L of A is also a RKHS, with kernelfunction kLλ := PLkλ.
For f ∈ A and g ∈ L we have fg ∈ L. Call this multiplicationoperator ML
f .
Ryan Hamilton Operator corona problems for function algebras
Our approach
Suppose f1, . . . , fn ∈ A and let
F := [f1, . . . , fn]; MF := [Mf1 , . . . ,Mfn ] : H(n) → H.
ThenM∗Fkλ = F (λi )
∗kλ
and the following are equivalent
MFM∗F =
n∑i=1
MfiM∗fi≥ c2IH
⟨(MFM
∗F − c2
)h, h⟩≥ 0, h ∈ H
[(〈F (λi )
∗,F (λj)∗〉 − c2
)〈kλi , kλj 〉
]ki ,j=1
≥ 0.
Ryan Hamilton Operator corona problems for function algebras
Our approach
Suppose f1, . . . , fn ∈ A and let
F := [f1, . . . , fn]; MF := [Mf1 , . . . ,Mfn ] : H(n) → H.
ThenM∗Fkλ = F (λi )
∗kλ
and the following are equivalent
MFM∗F =
n∑i=1
MfiM∗fi≥ c2IH
⟨(MFM
∗F − c2
)h, h⟩≥ 0, h ∈ H
[(〈F (λi )
∗,F (λj)∗〉 − c2
)〈kλi , kλj 〉
]ki ,j=1
≥ 0.
Ryan Hamilton Operator corona problems for function algebras
Our approach
Suppose f1, . . . , fn ∈ A and let
F := [f1, . . . , fn]; MF := [Mf1 , . . . ,Mfn ] : H(n) → H.
ThenM∗Fkλ = F (λi )
∗kλ
and the following are equivalent
MFM∗F =
n∑i=1
MfiM∗fi≥ c2IH
⟨(MFM
∗F − c2
)h, h⟩≥ 0, h ∈ H
[(〈F (λi )
∗,F (λj)∗〉 − c2
)〈kλi , kλj 〉
]ki ,j=1
≥ 0.
Ryan Hamilton Operator corona problems for function algebras
Our approach
Suppose f1, . . . , fn ∈ A and let
F := [f1, . . . , fn]; MF := [Mf1 , . . . ,Mfn ] : H(n) → H.
ThenM∗Fkλ = F (λi )
∗kλ
and the following are equivalent
MFM∗F =
n∑i=1
MfiM∗fi≥ c2IH
⟨(MFM
∗F − c2
)h, h⟩≥ 0, h ∈ H
[(〈F (λi )
∗,F (λj)∗〉 − c2
)〈kλi , kλj 〉
]ki ,j=1
≥ 0.
Ryan Hamilton Operator corona problems for function algebras
Our approach
Suppose f1, . . . , fn ∈ A and let
F := [f1, . . . , fn]; MF := [Mf1 , . . . ,Mfn ] : H(n) → H.
ThenM∗Fkλ = F (λi )
∗kλ
and the following are equivalent
MFM∗F =
n∑i=1
MfiM∗fi≥ c2IH
⟨(MFM
∗F − c2
)h, h⟩≥ 0, h ∈ H
[(〈F (λi )
∗,F (λj)∗〉 − c2
)〈kλi , kλj 〉
]ki ,j=1
≥ 0.
Ryan Hamilton Operator corona problems for function algebras
Our approach
The same observation is true for
n∑i=1
MLfi
(MLfi
)∗ ≥ c2IL
as well, with kL instead of k .
Ryan Hamilton Operator corona problems for function algebras
Our approach
Let E = λ1, . . . , λk ⊂ X .Suppose the condition[(
〈F (λi )∗,F (λj)
∗〉 − c2)〈kLλi , k
Lλj〉]ki ,j=1
≥ 0, L ∈ L (1)
implied the existence of MGE := [MgE1, . . . ,MgE
n]T such that
gEi ∈ A∑ni=1 fi (λj)g
Ei (λj) = 1 for j = 1 . . . k
‖MGE ‖ ≤ c−1.
Ryan Hamilton Operator corona problems for function algebras
Our approach
Let E = λ1, . . . , λk ⊂ X .Suppose the condition[(
〈F (λi )∗,F (λj)
∗〉 − c2)〈kLλi , k
Lλj〉]ki ,j=1
≥ 0, L ∈ L (1)
implied the existence of MGE := [MgE1, . . . ,MgE
n]T such that
gEi ∈ A
∑ni=1 fi (λj)g
Ei (λj) = 1 for j = 1 . . . k
‖MGE ‖ ≤ c−1.
Ryan Hamilton Operator corona problems for function algebras
Our approach
Let E = λ1, . . . , λk ⊂ X .Suppose the condition[(
〈F (λi )∗,F (λj)
∗〉 − c2)〈kLλi , k
Lλj〉]ki ,j=1
≥ 0, L ∈ L (1)
implied the existence of MGE := [MgE1, . . . ,MgE
n]T such that
gEi ∈ A∑ni=1 fi (λj)g
Ei (λj) = 1 for j = 1 . . . k
‖MGE ‖ ≤ c−1.
Ryan Hamilton Operator corona problems for function algebras
Our approach
Let E = λ1, . . . , λk ⊂ X .Suppose the condition[(
〈F (λi )∗,F (λj)
∗〉 − c2)〈kLλi , k
Lλj〉]ki ,j=1
≥ 0, L ∈ L (1)
implied the existence of MGE := [MgE1, . . . ,MgE
n]T such that
gEi ∈ A∑ni=1 fi (λj)g
Ei (λj) = 1 for j = 1 . . . k
‖MGE ‖ ≤ c−1.
Ryan Hamilton Operator corona problems for function algebras
Our approach
This solves the operator corona problem for A!
The set of all such GE are contained in a weak∗ compactsubset.
Point evaluation is weak∗-continuous for A.
Thus, the GE accumulate at some G which satisfies‖MG‖ ≤ c−1 and
∑ni=1 figi = 1 .
Ryan Hamilton Operator corona problems for function algebras
Tangential Interpolation
We have reduced the problem to a statement about interpolation.
Definition
Suppose λ1, . . . , λk ∈ X , v1, . . . , vk ∈ `2n and w1, . . . ,wk ∈ C.
A collection L of invariant subspaces for A is said to be atangential family if the following statement holds:There is a contractive column multiplier MG = [Mg1 , . . . ,Mgn ]T
with gi ∈ A such that 〈G (λi ), vi 〉Cn = wi for each i if and only if[(〈vi , vj〉Cn − wiwj
)〈kLλi , k
Lλj〉H]ki ,j=1
, L ∈ L
is positive semidefinite.
When F (λi )∗ = vi and wi = c, this is just the previous matrix.
Ryan Hamilton Operator corona problems for function algebras
Tangential Interpolation
We have reduced the problem to a statement about interpolation.
Definition
Suppose λ1, . . . , λk ∈ X , v1, . . . , vk ∈ `2n and w1, . . . ,wk ∈ C.
A collection L of invariant subspaces for A is said to be atangential family if the following statement holds:There is a contractive column multiplier MG = [Mg1 , . . . ,Mgn ]T
with gi ∈ A such that 〈G (λi ), vi 〉Cn = wi for each i if and only if[(〈vi , vj〉Cn − wiwj
)〈kLλi , k
Lλj〉H]ki ,j=1
, L ∈ L
is positive semidefinite.
When F (λi )∗ = vi and wi = c, this is just the previous matrix.
Ryan Hamilton Operator corona problems for function algebras
Tangential Interpolation
We have reduced the problem to a statement about interpolation.
Definition
Suppose λ1, . . . , λk ∈ X , v1, . . . , vk ∈ `2n and w1, . . . ,wk ∈ C.
A collection L of invariant subspaces for A is said to be atangential family if the following statement holds:
There is a contractive column multiplier MG = [Mg1 , . . . ,Mgn ]T
with gi ∈ A such that 〈G (λi ), vi 〉Cn = wi for each i if and only if[(〈vi , vj〉Cn − wiwj
)〈kLλi , k
Lλj〉H]ki ,j=1
, L ∈ L
is positive semidefinite.
When F (λi )∗ = vi and wi = c, this is just the previous matrix.
Ryan Hamilton Operator corona problems for function algebras
Tangential Interpolation
We have reduced the problem to a statement about interpolation.
Definition
Suppose λ1, . . . , λk ∈ X , v1, . . . , vk ∈ `2n and w1, . . . ,wk ∈ C.
A collection L of invariant subspaces for A is said to be atangential family if the following statement holds:There is a contractive column multiplier MG = [Mg1 , . . . ,Mgn ]T
with gi ∈ A such that 〈G (λi ), vi 〉Cn = wi for each i if and only if[(〈vi , vj〉Cn − wiwj
)〈kLλi , k
Lλj〉H]ki ,j=1
, L ∈ L
is positive semidefinite.
When F (λi )∗ = vi and wi = c, this is just the previous matrix.
Ryan Hamilton Operator corona problems for function algebras
Tangential Interpolation
We have reduced the problem to a statement about interpolation.
Definition
Suppose λ1, . . . , λk ∈ X , v1, . . . , vk ∈ `2n and w1, . . . ,wk ∈ C.
A collection L of invariant subspaces for A is said to be atangential family if the following statement holds:There is a contractive column multiplier MG = [Mg1 , . . . ,Mgn ]T
with gi ∈ A such that 〈G (λi ), vi 〉Cn = wi for each i if and only if[(〈vi , vj〉Cn − wiwj
)〈kLλi , k
Lλj〉H]ki ,j=1
, L ∈ L
is positive semidefinite.
When F (λi )∗ = vi and wi = c, this is just the previous matrix.
Ryan Hamilton Operator corona problems for function algebras
Tangential interpolation implies a solution
To summarize
Lemma
Suppose L is a tangential family for A. If f1, . . . , fn ∈ A satisfy
n∑i=1
MLfi
(MLfi
)∗ ≥ c2IL, L ∈ L
then there are g1, . . . , gn ∈ A such that
n∑i=1
figi = 1 and ‖[Mg1 , . . . ,Mgn ]T‖ ≤ c−1
Ryan Hamilton Operator corona problems for function algebras
Elementary spaces of operators
When does an algebra of multipliers A admit a tangential family?
Definition
A weak∗-closed subspace S of B(H) is said to be elementary ifevery ϕ ∈ S∗ with ‖ϕ‖ < 1 can be factored as
ϕ(A) = 〈Ax , y〉, A ∈ S
for some x , y ∈ H with ‖x‖‖y‖ < 1.
Ryan Hamilton Operator corona problems for function algebras
Elementary spaces of operators
Define the column space of A:C (A) := [Mg1 , . . . ,Mgn ]T : gi ∈ A ⊂ B(H(n),H)
Theorem
Suppose C (A) is elementary. Then A[h] : h ∈ H is a tangentialfamily for A.
Ryan Hamilton Operator corona problems for function algebras
Elementary spaces of operators
Define the column space of A:C (A) := [Mg1 , . . . ,Mgn ]T : gi ∈ A ⊂ B(H(n),H)
Theorem
Suppose C (A) is elementary. Then A[h] : h ∈ H is a tangentialfamily for A.
Ryan Hamilton Operator corona problems for function algebras
Proof.
Let J = G ∈ C (A) : 〈G (λi ), vi 〉Cn = 0 for i = 1, . . . , k.
If H ∈ C (A) is any column satisfying 〈H(xi ), vi 〉Cn = wi , thenH + G is also a solution for any G ∈ J .
We have a contractive solution if and only if dist(H, J) ≤ 1.
A standard distance argument shows that[(〈vi , vj〉 − wiwj) 〈kLλi , k
Lλj〉]ki ,j=1
≥ 0, L ∈ L
implies dist(H, J) ≤ 1 when C (A) is elementary.
Ryan Hamilton Operator corona problems for function algebras
Proof.
Let J = G ∈ C (A) : 〈G (λi ), vi 〉Cn = 0 for i = 1, . . . , k.If H ∈ C (A) is any column satisfying 〈H(xi ), vi 〉Cn = wi , thenH + G is also a solution for any G ∈ J .
We have a contractive solution if and only if dist(H, J) ≤ 1.
A standard distance argument shows that[(〈vi , vj〉 − wiwj) 〈kLλi , k
Lλj〉]ki ,j=1
≥ 0, L ∈ L
implies dist(H, J) ≤ 1 when C (A) is elementary.
Ryan Hamilton Operator corona problems for function algebras
Proof.
Let J = G ∈ C (A) : 〈G (λi ), vi 〉Cn = 0 for i = 1, . . . , k.If H ∈ C (A) is any column satisfying 〈H(xi ), vi 〉Cn = wi , thenH + G is also a solution for any G ∈ J .
We have a contractive solution if and only if dist(H, J) ≤ 1.
A standard distance argument shows that[(〈vi , vj〉 − wiwj) 〈kLλi , k
Lλj〉]ki ,j=1
≥ 0, L ∈ L
implies dist(H, J) ≤ 1 when C (A) is elementary.
Ryan Hamilton Operator corona problems for function algebras
Proof.
Let J = G ∈ C (A) : 〈G (λi ), vi 〉Cn = 0 for i = 1, . . . , k.If H ∈ C (A) is any column satisfying 〈H(xi ), vi 〉Cn = wi , thenH + G is also a solution for any G ∈ J .
We have a contractive solution if and only if dist(H, J) ≤ 1.
A standard distance argument shows that[(〈vi , vj〉 − wiwj) 〈kLλi , k
Lλj〉]ki ,j=1
≥ 0, L ∈ L
implies dist(H, J) ≤ 1 when C (A) is elementary.
Ryan Hamilton Operator corona problems for function algebras
Main Result
We say that a function h ∈ H2d is outer if M(H2
d)[h] = H2d .
Theorem
Suppose A ⊂M(H2d) is a unital, weak∗-closed subalgebra. Then
C (A) is elementary and every ϕ ∈ C (A)∗ can be factored as
ϕ(A) = 〈Ah, k〉
where h is an outer function.In other words L := A[h] : h outer is a tangential family for A.
Ryan Hamilton Operator corona problems for function algebras
Main result
Corollary
Suppose A is a unital, weak∗-closed subalgebra ofM(H2d) and
f1, . . . , fn ∈ A satisfy
n∑i=1
MLfi
(MLfi
)∗ ≥ c2IL
for every L of the form A[h] where h is an outer function. Thenthere are g1, . . . , gn ∈ A so that
n∑i=1
figi = 1 and ‖[Mg1 , . . . ,Mgn ]T‖ ≤ c−1
When A =M(H2d), this is the Ball-Trent-Vinnikov result. For
d = 1 it is the Raghupathi-Wick result.
Ryan Hamilton Operator corona problems for function algebras
Main result
Corollary
Suppose A is a unital, weak∗-closed subalgebra ofM(H2d) and
f1, . . . , fn ∈ A satisfy
n∑i=1
MLfi
(MLfi
)∗ ≥ c2IL
for every L of the form A[h] where h is an outer function. Thenthere are g1, . . . , gn ∈ A so that
n∑i=1
figi = 1 and ‖[Mg1 , . . . ,Mgn ]T‖ ≤ c−1
When A =M(H2d), this is the Ball-Trent-Vinnikov result. For
d = 1 it is the Raghupathi-Wick result.
Ryan Hamilton Operator corona problems for function algebras
Additional examples
For Ω ⊂ Cd recall the Bergman space L2a(Ω) and its multipliers
M(L2a(Ω)) = H∞(Ω).
Theorem (Bercovici 1987)
C (H∞(Ω)) is an elementary subspace of B(L2a(Ω), L2
a(Ω)⊗ `2n).
For h ∈ L2a(Ω), the cyclic subspace generated by h may be identified
with L2a(Ω, |h|2dµ).
A sufficient condition to solve the Toeplitz corona problem for thesealgebras is
MνF (M∗F )ν ≥ c2Iν
for the absolutely continuous measures ν = |h|2µ on Ω.
Ryan Hamilton Operator corona problems for function algebras
Additional examples
For Ω ⊂ Cd recall the Bergman space L2a(Ω) and its multipliers
M(L2a(Ω)) = H∞(Ω).
Theorem (Bercovici 1987)
C (H∞(Ω)) is an elementary subspace of B(L2a(Ω), L2
a(Ω)⊗ `2n).
For h ∈ L2a(Ω), the cyclic subspace generated by h may be identified
with L2a(Ω, |h|2dµ).
A sufficient condition to solve the Toeplitz corona problem for thesealgebras is
MνF (M∗F )ν ≥ c2Iν
for the absolutely continuous measures ν = |h|2µ on Ω.
Ryan Hamilton Operator corona problems for function algebras
Additional examples
For Ω ⊂ Cd recall the Bergman space L2a(Ω) and its multipliers
M(L2a(Ω)) = H∞(Ω).
Theorem (Bercovici 1987)
C (H∞(Ω)) is an elementary subspace of B(L2a(Ω), L2
a(Ω)⊗ `2n).
For h ∈ L2a(Ω), the cyclic subspace generated by h may be identified
with L2a(Ω, |h|2dµ).
A sufficient condition to solve the Toeplitz corona problem for thesealgebras is
MνF (M∗F )ν ≥ c2Iν
for the absolutely continuous measures ν = |h|2µ on Ω.
Ryan Hamilton Operator corona problems for function algebras